On $C$-embedded subspaces of the Sorgenfrey plane

TL;DR

This paper investigates the properties of $C$-embedded subsets within the Sorgenfrey plane, establishing their Baire category characteristics and equivalences among various embedding conditions for specific subspaces.

Contribution

It proves that $C^*$-embedded subsets of the Sorgenfrey plane are hereditarily Baire and characterizes $C$-embedded subspaces of a particular line segment as equivalent to several topological conditions.

Findings

$C^*$-embedded subsets are hereditarily Baire

Equivalence of $C$-embedded and $C^*$-embedded for certain subspaces

Characterization of subspaces as countable $G_\delta$ and functionally closed

Abstract

We prove that every $C^*$-embedded subset of $\ss$ is a hereditarily Baire subspace of $\mathbb R^2$. We also show that for a subspace $E\subseteq\{(x,-x):x\in\mathbb R\}$ of the Sorgenfrey plane $\mathbb S^2$ the following conditions are equivalent: (i) $E$ is $C$-embedded in $\mathbb S^2$; (ii) $E$ is $C^*$-embedded in $\mathbb S^2$; (iii) $E$ is a countable $G_\delta$-subspace of $\mathbb R^2$ and (iv) $E$ is a countable functionally closed subspace of $\mathbb S^2$.